February 21, 2022
(Estes et al. 1974)
> … the rush for the otters’ “soft gold” was a predictable boom and bust … a cautionary example of unsustainable resource use, and a socioeconomic driver of Western — mainly American — involvement in the Pacific region, (Loshbaugh 2021)
art by John Livingston
… a sea otter hunter who was “lazy, spiteful, malicious or disregarded the teaching of the elders” would find his prey “cavorting around his baidarka … teas[ing] and splash[ing] him with water.” Venianimov
Remnant populations from Aleutian Islands … released in OR, WA, BC and SE-AK 1969 – 1972.
1970: 60 otters
2010’s: over 1000
\[\huge N_t\]
\[\Large N_t = N_{t - \Delta t} + \Delta N\]
slight rearrangement:
\[\Large N_{t+1} = N_t + \Delta N\] For now, \(\Delta t = 1\), i.e. it’s the discrete unit that we measure population change. VERY TYPICALLY - whether because of biology or field seasons: \[\Delta t = 1\,\, \textrm{year}\].
This is a closed population …
The number of Births and Deaths is proportional to N.
\[\Large N_{t+1} = N_t + bN_t - dN_t\] What does that mean?
Define \[r_0 = b - d\]
(Gotelli calls this: discrete growth factor, Neal calls this: something else).
\[N_{t+1} = N_t + r_0 N_t\] \[N_{t+1} = (1 + r_0) N_t\]
\[N_{t+1} = \lambda N_t\]
If \(d > b\), \(\lambda < 1\). If \(b > d\), \(\lambda > 1\).
(Gotelli calls this finite rate of increase, Neal calls this: something else).
\[N_{t+1} = \lambda(N_t)\] \[N_{t+2} = \lambda(N_{t+1}) = \lambda^2 N_t\] \[N_{t+3} = \lambda^3 N_t\]
Solution:
\[\large N_{t+y} = \lambda^y N_t\] or
\[\huge N_t = \lambda^t N_0\]
Let’s do some trickery, starting with: \[N_{t+1} = (1 + r_0) N_t\] \[N_{t+1} - N_t = r_0 N_t\] \[N_{t+\Delta t} - N_t = r_{\Delta t} N_t\]
\[\lim_{\Delta t \to 0} {\Delta N \over \Delta t} = \lim_{\Delta t \to 0} {r_{\Delta t} \over \Delta t} N\]
Magically define: \({r_\Delta \over \Delta t} = r\) and rewrite \(\Delta\) as \(d\):
\[\large {dN \over dt} = r N\]
Calculate:
\(\begin{align} {1\over N} dN &= r dt \\ \int_{t' = t_0}^t {1 \over N(t)} dN &= \int_{t' = t_0}^t r dt \\ \log(N) &= rt + C_0 \\ N &= e^{rt + C_0} \\ \end{align}\)
Solution:
Plug in: \[N(0) = N_0\\\] \[ \large N(t) = N_0 e^{rt}\]
Important fact: The exponential function is the ONLY function whose derivative is proportional to itself.
Difference equations \[N_{t+\Delta t} - N_t = \lambda_{\Delta t} N_t\]
think of absolute change
Pros:
Cons:
Differential equations \[{dN \over dt} = r N\]
think of rates (change/time).
Pros
Cons
The amazing thing is, if you have an equation “solved”, you only need 2 points on the curve to compute.
Let’s use the discrete equation:
\[N_{t+y} = \lambda^y N_t\] \[1000 = 60 \times \lambda^{40}\] \[16.7 = \lambda^{40}\] \[2.81 = 40 \times \log (\lambda)\] \[0.07025 = \log (\lambda)\] \[\lambda = \exp(0.07025) = 1.0728\]
i.e. population increase about \[7.28\%/year\].
Let’s use the discrete equation:
\[1000 = 60 \times e^{40\,r}\] \[16.7 = e^{40 \, r}\] \[2.81 = 40 \times r\] \[r = 0.07025\] Intrinsic growth rate = 0.07025.
solve for \(t_d\)
Continuous:
\[2N = N e^{rt_d}\] \[\log(2) = rt_d\] \[t_d = \log(2)/r\]
Discrete:
\[2N = N \lambda^{t_d}\] \[\log(2) = \log(\lambda) t_d\] \[t_d = \log(2)/\log(\lambda)\]